Tangents and Normals - Rajasthan Board (RBSE) STD 12 Science MATHS Questions

Q 1M.C.Q (1 Marks)1 Mark

The angle of intersection of the curves $xy = a^2$ and $x^2 - y^2 = 2a^{2 }$ is:

A
$0^\circ$
B
$45^\circ$
✓
$90^\circ$
D
None of these.

Answer: C.

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Q 2M.C.Q (1 Marks)1 Mark

If the curve $ay + x^2 = 7$ and $x^3 = y$ cut orthogonally at $(1, 1)$, then a is equal to:

A
$1$
B
$-6$
✓
$6$
D
$0$

Answer: C.

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Q 3M.C.Q (1 Marks)1 Mark

The line $y = mx + 1$ is a tangent to the curve $y^2 = 4x,$ if the value of $m$ is:

✓
$1$
B
$2$
C
$3$
D
$\frac{1}{2}$

Answer: A.

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Q 4M.C.Q (1 Marks)1 Mark

The point on the curve $y = 6x - x^2$ at which the tangent to the curve is inclined at $\frac{\pi}{4}$ to the line $x + y = 0$ is:

A
$(-3,-27)$
✓
$(3,9)$
C
$\Big(\frac{7}{2},\frac{35}{4}\Big)$
D
$(0,0)$

Answer: B.

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Q 5M.C.Q (1 Marks)1 Mark

The point at the curve $y = 12x - x^2$ where the slope of the tangent is zero will be:

A
$(0, 0)$
B
$(2, 16)$
C
$(3, 9)$
✓
None of these.

Answer: D.

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Q 62 Marks Questions2 Marks

Write the value of $\Big(\frac{\text{dy}}{\text{dx}}\Big),$ if the normal to the curve y = f(x) at (x, y) is parallel to y-axis.

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Q 72 Marks Questions2 Marks

If the tangent to a curve at a point (x, y) is equally inclined to the co-ordinates axes then write the value of $\frac{\text{dy}}{\text{dx}}.$

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Q 82 Marks Questions2 Marks

Write the equation of the tangent drawn to the curve $\text{y}=\sin\text{x}$ at the point (0, 0).

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Q 92 Marks Questions2 Marks

Write the slop of the normal to the curve $\text{y}=\frac{1}{\text{x}}$ at the point $\Big(3,\frac{1}{3}\Big).$

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Q 102 Marks Questions2 Marks

If the tangent line at a point (x, y) on the curve y = f(x) is parallel to y-axis, find the value of $\frac{\text{dy}}{\text{dx}}.$

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Q 113 Marks Question3 Marks

Find the slope of the normal at the point 't' on the curve $\text{x}=\frac{1}{\text{t}},\text{y}=\text{t}.$

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Q 123 Marks Question3 Marks

Find the point on the curve $y = x^2$ where the slope of the tangent is equal to the $x-$coordinate of the point.

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Q 133 Marks Question3 Marks

Write the coordinates of the point at which the tangent to the curve $y = 2x^2 - x + 1$ is parallel to the line $y = 3x + 9.$

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Q 143 Marks Question3 Marks

Write the angle between the curves $y = e^{-x}$ and $y = e^x$ at their point of intersections.

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Q 153 Marks Question3 Marks

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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Q 165 Marks Questions5 Marks

Show that the curve $\frac{\text{x}^2}{\text{a}^2+\lambda_1}+\frac{\text{y}^2}{\text{b}^2+\lambda_1}=1$ and $\frac{\text{x}^2}{\text{a}^2+\lambda_2}+\frac{\text{y}^2}{\text{b}^2+\lambda_2}=1$ intersect at right angles.

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Q 175 Marks Questions5 Marks

Find the equations of the tangent and the normal to the following curves at the indicated points.
$\text{x}^{\frac{2}{3}}+\text{y}^{\frac{2}{3}}=2\text{ at }(1,1)$

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Q 185 Marks Questions5 Marks

Find the point on the curve $y = x^2 - 2x + 3,$ where the tangent is parallel to $x-$axis.

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Q 195 Marks Questions5 Marks

Find the points on the curve $x^2 + y^2 = 13,$ the tangent at each one of which is parallel to the line $2x + 3y = 7.$

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Q 205 Marks Questions5 Marks

Find the point on the curve $\frac{\text{x}^2}{9}+\frac{\text{y}^2}{16}=1$ at which the tangent are: