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Question

The angle at which the curve y=x2 and the curve x=53cost,y=54sint intersect is

A
tan1241
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B
tan1412
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C
tan1241
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D
2tan1412
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Solution

The correct option is A tan1412
Given
y=x2...(i)
x=53cost;y=54sint...(ii)
Which is parametric equation, we change this equation is caresian equation as follows
cost=35x;sint=45y
On Squaring and adding both i.e., cost and sint we get
925x2+1625y2=cos2t+sin2t
9x2+16y2=25....(iii)[cos2θ+sin2θ=1]
The intersection points at Eq.(i) and (iii) are (1,1) and (1,1)
Now, slope of tangent of Eq.(i) at point (1,1) is
m1=dydx=2xm1=dydx(1,1)=2
And slope of tangent of Eq. (iii) at point (1,1) is
m2=dydx=916
Angle at point of intersection of Eqs. (i) and (iii) we get
θ1=tan1m1m21+m1m2=tan1412
similarly, slope of tangent of Eq. (i) at point (1,1)
m1=dydx(1,1)=2
And slope of tangent of Eq. (iii) at point (1,1)
m2=dydx=916
Angle at point of intersection of Eqs. (i) and (iii) we get
θ2=tan1∣ ∣ ∣ ∣291611816∣ ∣ ∣ ∣=tan1412

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