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APPLICATION OF DERIVATIVES

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Question: If the tangent line at a point (x, y) on the curve y = f(x) is parallel to the x-axis, then write the value of dy/dx.
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Prove that the curves \( xy = 4 \) and \( x^2 + y^2 = 8 \) touch each other.
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Question: Find the equations of the tangent and the normal to the curve y² = 4ax at (x₁, y₁) at indicated points.
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5
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Find all the tangents to the curve \( y = \cos(x + y) \), \(-2\pi \leq x \leq 2\pi\) that are parallel to the line \( x + 2y = 0 \).
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14
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Question: The equation of the tangent at (2, 3) on the curve \( y^2 = ax^3 + b \) is \( y = 4x - 5 \). Find the values of \( a \) and \( b \).
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Question: Write the equation of the normal to the curve \( y = \cos x \) at \( (0, 1) \).
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Question: Find the equation of tangent and normal to the curve \( x = 2at^2(1 + t^2), y = 2at^3(1 + t^2) \) at the point for which \( t = 12 \).
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5
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Question: Find the equations of the normal to the curve \(xy = c^2\) at \((ct, c/t)\) on it.
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5
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Question: Find the equations of the tangent and the normal to the curve y = x^2 + 4x + 1 at x = 3 at the indicated points
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23
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Find the equation of the tangent to the curve y = −5x² + 6x + 7 at the point (1/2, 35/4).
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Question: Show that the tangents to the curve \( y = 7x^3 + 11 \) at the points \( x = 2 \) and \( x = -2 \) are parallel.
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14
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Question: Find the points on the curve \( x^2/9 + y^2/16 = 1 \) at which the tangents are parallel to the x-axis and y-axis.
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9
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Question: Find the point on the curve \( y = 3x^2 + 4 \) at which the tangent is perpendicular to the line whose slope is \(-16\).
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Question: Prove that the tangents to the curve y = x² − 5x + 6 at the points (2, 0) and (3, 0) are at right angles.
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Determine the quadratic curve \( y = f(x) \) if it touches the line \( y = x \) at the point \( x = 1 \) and passes through the point \( (-1,0) \).
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The normal to the curve x² = 4y passing through (1, 2) is (A) x + y = 3 (B) xy = 3 (C) x + y = 1 (D) xy = 1
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The normal at the point (1,1) on the curve 2y + x² = 3 is (A) x + y = 0 (B) xy = 0 (C) x + y + 1 = 0 (D) xy = 0
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3
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The line y = mx + 1 is a tangent to the curve y² = 4x, if the value of m is (a) 1 (b) 2 (c) 3 (d) 1/2
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13
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The slope of the tangent to the curve x = t²+3t−8, y = 2t²−2t−5 at the point (2,−1) is (a) 22/7 (b) 6/7 (c) 7/6 (d) −6/7
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7
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The point on the curve 9y² = x³, where the normal to the curve makes equal intercepts with the axes is (a) (4, ±8/3) (b) (-4, 8/3) (c) (-4, -8/3) (d)
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