The length of the chord of the ellipse 4x2+2y2=1, whose mid-point is (1,21), is:
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Step-by-Step Solution
Step 1: Equation of the chord with a given midpoint
The equation of a chord of an ellipse with a given midpoint (x1,y1) can be found using the formula T=S1. Here, T represents the tangent equation at (x1,y1) and S1 represents the value of the ellipse equation when (x, y) is replaced by (x1,y1). The given ellipse is 4x2+2y2=1, and the midpoint is (1,21).
Step 2: Calculate T and S1
Substitute x1=1 and y1=21 into the expressions for T and S1. The equation for T is 4xx1+2yy1−1, and for S1 it is 4x12+2y12−1.
Step 3: Simplify the chord equation
Simplify both sides of the equation. The left side becomes 4x+4y−1. The right side becomes 41+81−1=82+1−8=−85. So, 4x+4y−1=−85. Adding 1 to both sides gives 4x+4y=1−85=83. Multiplying by 8 gives 2x+2y=3. This is the equation of the chord.
Step 4: Find intersection points of the chord and ellipse
To find the length of the chord, we need its endpoints. We can find these by solving the equation of the chord (2x+2y=3⟹y=23−2x) simultaneously with the equation of the ellipse 4x2+2y2=1. Substitute the expression for y from the chord equation into the ellipse equation.
Step 5: Solve for x-coordinates of intersection points
Simplify the equation: 4x2+89−12x+4x2=1. Multiply by 8 to clear denominators: 2x2+9−12x+4x2=8. Rearrange into a quadratic equation: 6x2−12x+1=0. Let the roots be x1 and x2.
Step 6: Solve for y-coordinates of intersection points
Similarly, we can express x in terms of y from the chord equation: x=23−2y. Substitute this into the ellipse equation: 4(23−2y)2+2y2=1. This simplifies to 6y2−6y+1=0. Let the roots be y1 and y2.
Step 7: Calculate the length of the chord
The length of the chord L is given by the distance formula. For the quadratic equation ax2+bx+c=0, (x2−x1)2=(x1+x2)2−4x1x2=(−b/a)2−4(c/a). For 6x2−12x+1=0, x1+x2=12/6=2 and x1x2=1/6. So (x2−x1)2=(2)2−4(1/6)=4−2/3=10/3. For 6y2−6y+1=0, y1+y2=6/6=1 and y1y2=1/6. So (y2−y1)2=(1)2−4(1/6)=1−2/3=1/3. Therefore, L=310+31=311.